Section 17–1 The geometry of space-time

(Analogy of space-time / Paths in space-time / Axes of space-time)

In this section, Feynman discusses the geometry of space-time from the perspectives of an analogy, possible paths of an object, and the axes of a space-time diagram.

1. Analogy of space-time:

“An analogy is useful: When we look at an object, there is an obvious thing we might call the ‘apparent width,’ and another we might call the ‘depth’ (Feynman et al., 1963, section 17–1 The geometry of space-time).”

Feynman explains the concept of space-time using the “apparent width” and “apparent depth” of an object that are not its fundamental properties. This is because we get a different width and a different depth if we view the object from a different angle. One may also say that a given depth is a kind of “mixture” of all depth and all width that is similar to the two equations: x′ = xcosθ + ysinθ, y′ = ycosθ − xsinθ. Importantly, the analogy of space-time has its limitations. That is, there are no absolute space and absolute time that are in contrast to relative space-time. In general, the concepts of space and time are interrelated such that there is no “space without time” or “time without space” from the perspectives of observers in different reference frames.

Feynman elaborates that depth and width are just two different aspects of the same thing. Historically, in Minkowski’s (1907) words, “[f]rom now onwards space by itself and time by itself will recede completely to become mere shadows and only a type of union of the two will still stand independently on its own (p. 111).” Minkowski also explains that there is an infinite number of spaces analogously as there is an infinite number of planes in three-dimensional space. In a sense, time and space are analogous to the extent that the space-time interval can be expressed as Sa_i² (to be discussed in the next section). On the other hand, the temporal dimension is different from the spatial dimension if we express the space-time interval as +(ct)²–x² in which they can be distinguished by the different signs.

2. Paths in space-time:

“If the particle is standing still, then it has a certain x, and as time goes on, it has the same x, the same x, the same x; so its “path” is a line that runs parallel to the t-axis (Feynman et al., 1963, section 17–1 The geometry of space-time).”

According to Feynman, the geometrical entity in which the “blobs” exist by occupying a range of position and occurring during a certain amount of time is called space-time. In addition, a given point (x, y, z, t) in space-time is called an event. Alternatively, we may explain that a Minkowski space-time diagram is a two-dimensional graph that depicts a sequence of events (or world-points) corresponding to the history of an object. We may define a world-point (or an event) as something that occurs at a point in space-time in which it has a definite location and a definite time. Furthermore, the path of an object that is shown in the space-time diagram is also called the world line (Minkowski, 1907).

Feynman describes a stationary object in a space-time diagram as having a certain x, and as time goes on, it has the same x whereby its “path” is a line that is parallel to the t-axis. On the other hand, it is impossible to have a path that is parallel to the x-axis which implies the object exists in everywhere at the same time. (We may interpret the x-axis as a “line of now,” whereas the t-axis as a “line of here.”) However, one may clarify that the t-axis of a space-time diagram is scaled with the speed of light c and labeled by ct such that a light ray’s path is shown as a bisector of the angle between the two axes. This is related to the amazing fact that the Lorentz transformations preserve the quadratic form c²t²-x² and thus, they can be regarded geometrically as a rotation in a four-dimensional space. Poincare should be credited for his geometric interpretation of Lorentz transformations of space-time.

3. Axes of space-time:

“In fact, although we shall not emphasize this point, it turns out that a man who is moving has to use a set of axes which are inclined equally to the light ray, using a special kind of projection parallel to the x′- and t′-axes (Feynman et al., 1963, section 17–1 The geometry of space-time).”

A given event can be represented using different axes of x′ and t′, but it is not exactly the same mathematical transformation as shown by the two equations: x′ = xcosθ + ysinθ, y′ = ycosθ − xsinθ. Curiously, Feynman mentions that it is not impossible that the algebraic quantities could be written in terms of cosine and sine, and adds that “but actually they cannot.” However, we can relate x and x′ using the following two forms of equations: x = ax′ + bt′ and x′ = ax - bt. More important, we should not assume one unit of distance on different axes of x and x′ must have the same length (French, 1971). We need to draw the rectangular hyperbola (or calibration curve) as defined by x²−(ct)² =1 to calibrate the unit distance from different axes.

Feynman explains that it is not really possible to think of space-time as a real, ordinary geometry because of a difference in sign in x²−(ct)². It is surprising that he prefers not to deal with the space-time geometry and even explains that this approach does not help much because it is easier to work with the equations. To be precise, the Lorentz transformations equations can be related to hyperbolic geometry and expressed in terms of hyperbolic functions of a real angle. Furthermore, we may draw space-time diagrams to help students develop a geometric understanding of the “time dilation” and “length contraction” without using mathematical equations. Students may also find the graphical presentations of relativity concepts insightful.

Questions for discussion:

1. How would you provide an analogy of space-time?

2. How would you describe different paths of an object in space-time diagrams?

3. How would you explain the axes of x′ and t′ in space-time diagrams?

The moral of the lesson: a space-diagram is related to the amazing fact that the Lorentz transformations preserve the quadratic form c²t²-x² and the transformation of coordinates can be regarded geometrically as a rotation in a four-dimensional space.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Minkowski, H. (1907). Space and Time. In Petkov, V., Ed. Minkowski’s Papers on Relativity. Moscu: Minkowski Institute Press.